Construction algorithms for higher order polynomial lattice rules

Abstract

Higher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. The existence of higher order polynomial lattice point sets of “good” quality has recently been established, but their construction was not addressed. We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time–under certain conditions on the weights–(strong) polynomial tractability. Combining this approach with a sieve-type algorithm yields higher order polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree. Higher order Korobov polynomial lattice rules achieve analogous results.